Solitons in quasi-one-dimensional Bose-Einstein condensates with competing dipolar and local interactions

We study families of one-dimensional matter-wave bright solitons supported by the competition of contact and dipole-dipole (DD) interactions of opposite signs. Soliton families are found, and their stability is investigated in the free space and in the presence of an optical lattice (OL). Free-space solitons may exist with an arbitrarily weak local attraction if the strength of the DD repulsion is fixed. In the case of the DD attraction, solitons do not exist beyond a maximum value of the local-repulsion strength. In the system which includes the OL, a stability region for subfundamental solitons is found in the second finite band gap. For the existence of gap solitons (GSs) under the attractive DD interaction, the contact repulsion must be strong enough. In the opposite case of the DD repulsion, GSs exist if the contact attraction is not too strong. Collisions between solitons in the free space are studied too. In the case of the local attraction, they merge or pass through each other at small and large velocities, respectively. In the presence of the local repulsion, slowly moving solitons bounce from each other.

Figure 1

(Color online) The solid and dashed lines show the CO kernel, with $y_c={\pi }^{-1/2}$ and its SMA counterpart, which are based on Eqs. (2, 3), respectively.

Figure 2

(Color online) (a) Profiles of stable solitons in the model with attractive local and repulsive nonlocal interactions. The solitons were found in the numerical form with resolution $\Delta x=\pi /40$, which yields accurate results for $\left|g_c\right|>0.9$. The peak gets sharper as $\left|g_c\right|$ decreases. (b) Profiles of solitons in the model with repulsive local and attractive nonlocal interactions, obtained with $\Delta x=\pi /10$. The compactonlike solution corresponding to $\left|g_c\right|=3.6$ is close to the critical point, beyond which (at larger $\left|g_c\right|$) solitons do not exist, irrespective of the value of $\Delta x$. In both panels, all solitons pertain to $\mu =-1$.

Figure 3

(Color online) (a) The norm of solitons in the free space, in the case of attractive local and repulsive nonlocal interactions, versus the chemical potential. (b) The width of the solitons, defined as per Eq. (5), versus the strength of the local attraction. In panel (b), the plots are shown for $W>\Delta x=\pi /40$: recall that obtaining numerical results for smaller $W$ requires using smaller $\Delta x$.

Figure 4

(Color online) The norm and width of fundamental solitons in the free space, in the case of attractive nonlocal and repulsive local interactions, versus the strength of the local repulsion.

Figure 5

(Color online) The norm and width of fundamental solitons in the free space, in the case of attractive nonlocal and repulsive local interactions, versus the chemical potential.

Figure 6

(Color online) The profile of a stable dark soliton in the model with attractive local and repulsive nonlocal interactions for $g_c=5$, $g_d=-1$, and $\mu =1$. The soliton was found in the numerical form with resolution $\Delta x=\pi /10$.

Figure 7

(Color online) Unshaded and light-gray-shaded areas represent existence regions for solitons in the semi-infinite and two lowest finite gaps in the model combining the optical lattice with competing attractive local and repulsive nonlocal interactions. Dark-gray shading covers Bloch bands, where solitons do not exist. In the gray-shaded parts of the finite band gaps (intermediate between light and dark gray), the solitons do not exist either. In the white part of the second band gap on the left side of the vertical dashed line, only SFSs exist. They are stable between the red (solid) line and the gray area. On the right side of the dashed line, stable fundamental solitons exist in the white region, while unstable SFSs exist both in the white and light-gray areas.

Figure 8

(Color online) $N\left(\mu \right)$ curves for gap solitons in the model including the optical lattice, local attraction, and nonlocal repulsion. The strength of the local attraction is fixed to $g_c=-1$.

Figure 9

(Color online) Profiles of stable solitons labeled A–F in Fig. 8. Additionally, panels G–J display examples of stable regular solitons found in the semi-infinite gap. The parameters are ($g_c=-1.5$, $\mu =1$) and ($g_c=-1$, $\mu =3$ and 5.5) for the solitons in the semi-infinite gap and first and second finite band gaps, respectively.

Figure 10

(Color online) The existence range for solitons in the semi-infinite and two lowest finite gaps in the model with the optical lattice and competing local repulsive and nonlocal attractive interactions. The notation is the same as in Fig. 7.

Figure 11

(Color online) $N\left(\mu \right)$ curves for regular (left panel) and gap (right panel) solitons in the model with the optical lattice, local repulsion, and nonlocal attraction. The strength of the local repulsion is fixed to (a) $g_c=1$ and (b) $g_c=5$.

Figure 12

(Color online) Soliton profiles at points A–J in Fig. 11. The parameters are ($g_c=1$, $\mu =1$) and ($g_c=5$, $\mu =3$ and 5.5) for the solitons in the semi-infinite gap and first and second finite band gaps, respectively.

Figure 13

(Color online) Soliton-soliton collisions in the free space $\left(ϵ=0\right)$ for $g_c=1$, $g_d=-1$, and $\mu =-1$. The initial velocities are (a) $c=\pm 0.2942$ and (b) $c=\pm 0.9651$.

Figure 14

(Color online) Soliton collisions at $g_c=-5$, $g_d=1$, $\mu =-1$, and $ϵ=0$. Initial velocities are (a) $c=\pm 0.0990$, (b) $\pm 0.1984$, and (c) $\pm 0.4969$. These plots were generated using the mesh with $\Delta x=0.1$.

Figure 15

(Color online) Critical values of the collision velocity versus the strength of the contact attraction or repulsion for (a) $g_d=1$ and (b) $g_d=-1$. In both cases, $\mu =-1$ and $ϵ=0$.